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At first glance it appears that one way to get around the problem of choosing the proper strucutre is to let the structure be an arbitrary nonlinear time-varying system. In other words, the class of structure is chosen to be so large that every possible system will be included in it. The difficulty is that there is no convenient tool, such as the convolution integral, to express the output of a nonlinear system in terms of its input. This means that there is no convenient way to investigate all possible systems by using a structured approach.
The alternative to the structured approach is a nonstructured approach. Here we refuse to make any a priori guesses about what structure the processor should have. We establish a criterion, solve the problem, and implement whatever processing procedure is indicated.
A simple example of the nonstructured approach can be obtained by modifying Example 2. Instead of assigning characteristics to the device, we denote the estimate by y(t). Letting
Ø) é. ÅØt) - *(0]2}, (37)
we solve for the y(t) that is obtained from r(t) in any manner to minimize The obvious advantage is that if we can solve the problem we know that our answer, is with respect to the chosen criterion, the best processor of all possible processors. The obvious disadvantage is that we must completely characterize all the signals, channels, and noises that enter into the problem. Fortunately, it turns out that there are a large number of problems of practical importance in which this complete characterization is possible. Throughout both books we shall emphasize the nonstructured approach.
Our discussion up to this point has developed the topical and logical basis of these books. We now discuss the actual organization.
The material covered in this book and Volume II can be divided into five parts. The first can be labeled Background and consists of Chapters 2 and 3. In Chapter 2 \ye develop in detail a topic that we call Classical Detection and Estimation Theory. Here we deal with problems in which
16 1.3 Organization
the observations are sets of random variables instead of random waveforms. The theory needed to solve problems of this type has been studied by statisticians for many years. We therefore use the adjective classical to describe it. The purpose of the chapter is twofold: first, to derive all the basic statistical results we need in the remainder of the chapters; second, to provide a general background in detection and estimation theory that can be extended into various areas that we do not discuss in detail. To accomplish the second purpose we keep the discussion as general as possible. We consider in detail the binary and M-ary hypothesis testing problem, the problem of estimating random and nonrandom variables, and the composite hypothesis testing problem. Two more specialized topics, the general Gaussian problem and performance bounds on binary tests, are developed as background for specific problems we shall encounter later.
The next step is to bridge the gap between the classical case and the waveform problems discussed in Section 1.1. Chapter 3 develops the necessary techniques. The key to the transition is a suitable method for characterizing random processes. When the observation interval is finite, the most useful characterization is by a series expansion of the random process which is a generalization of the conventional Fourier series. When the observation interval is infinite, a transform characterization, which is a generalization of the usual Fourier transform, is needed. In the process of developing these characterizations, we encounter integral equations and we digress briefly to develop methods of solution. Just as in Chapter 2, our discussion is general and provides background for other areas of application.
With these two chapters in the first part as background, we are prepared to work our way through the hierarchy of problems outlined in Figs. 1.4,
1.7, and 1.10. The second part of the book (Chapter 4) can be labeled Elementary Detection and Estimation Theory. Here we develop the first two levels described in Section 1.1. (This material corresponds to the upper two levels in Figs. 1.4 and 1.7.) We begin by looking at the simple binary digital communication system described in Fig. 1.1 and then proceed to more complicated problems in the communications, radar, and sonar area involving M-àòó communication, random phase channels, random amplitude and phase channels, and colored noise interference. By exploiting the parallel nature of the estimation problem, results are obtained easily for the estimation problem outlined in Fig. 1.5 and other more complex systems. The extension of the results to include the multiple channel (e.g., frequency diversity systems or arrays) and multiple parameter (e.g., range and Doppler) problems completes our discussion. The results in this chapter arp fundamental to the understanding of modern communication and radar/sonar systems.